Split Representations and Bubble Resummation for Massive de Sitter Correlators

TL;DR

This work develops a split representation combined with a Källén-Lehmann spectral framework to factorize and resum multi-loop scalar diagrams in de Sitter momentum space within the Schwinger-Keldysh formalism. By exploiting a conserved spectral parameter, the authors extend flat-space resummation techniques to de Sitter, enabling analytic control over bubble-chain contributions in a large-$N$ model and revealing direct access to cosmological collider signals through the spectral function. In particular, they derive explicit constructions of the de Sitter harmonic function $\Omega_\nu$, prove a Delta-completeness identity, and demonstrate factorization and renormalization of bubble chains, including closed-form results in $d=2$ and partial results in $d=3$. The analysis clarifies the background (EFT) versus signal (cosmological collider) components of the 1-loop and resummed correlators, discusses the pole-flow under coupling renormalization, and highlights positivity and unitarity constraints for positive couplings with implications for more general exchange diagrams. Overall, the method provides a practical, non-perturbative avenue to extract oscillatory collider signals and EFT backgrounds from de Sitter correlators, with potential extensions to other exchange topologies and higher dimensions.

Abstract

We combine spectral- and split representations to factorize multi-loop momentum space diagrams, in the Schwinger-Keldysh formulation for cosmological correlators, with massive scalars in the loop. This allows us to extend the resummation of loop contributions from flat to de Sitter space. Furthermore, in our split representation the signal part of the correlators can be identified directly on the integrand level from the spectral function. We apply this to describe the non-perturbative flow of the EFT background and the cosmological collider signals in a large-N model.

Figures (7)

• Figure 1: Numerical comparison of the contact diagram via split representations where we truncate the sum over hypergeometric functions after $N=1,2,5,10$ terms, to the exact result from direct integration. For the evaluation, we fixed the conformal momentum ratio $r_2= 0.5$ and varied $r_1$. The two plots show the same curve with different scaling on the horizontal axis. We will use the logarithmic scaling later for displays of cosmological collider signals. The linear scaling in the right plot highlights the fact that the series representation approximates the exact solution very well even for a small number of terms until it gets close to the diagonal where $r_1=r_2$. The results are plotted in units of $\lambda \eta_*^4/(16k_1k_2k_3k_4k_s)$.
• Figure 2: Spectral function in $d=2$ dimensions for different masses $\nu$ and $\nu'$. From these numerical studies we can see that the spectral function is indeed positive along the real axis.
• Figure 3: Plot of the signal part of the 1-loop correction to the four-point function, i.e. the sum of $\mathcal{I}_{\rangle\!\bigcirc\!\langle,NS}^\nu$ and $\mathcal{I}_{\rangle\!\bigcirc\!\langle,LS}^\nu$, in $d=3$ dimensions for four different masses $\nu$. We fixed $r_2 = 0.5$ and varied $r_1\in(10^{-5}, r_2)$. The sums in the expressions of the signals were truncated after 30 terms. For better visibility of the oscillations we scaled the result by $(r_1/r_2)^{-3/2}$ so that only the $r_1$-dependence with power $\pm 2\nu$ survives in the squeezed limit, as well as a numerical factor of $10^6$.
• Figure 4: Pole structure of the 1-loop spectral function $\Sigma_\nu(\tilde{\nu}')$ for $\tilde{\nu}=2.0$ in the complex plane which has poles at $\pm i(\frac{d}{2}+2m)$ and $\pm i(\frac{d}{2}+2m\pm \nu),\,m\in\mathbb{N}$. The poles at $\pm i(\frac{d}{2}+2m)$ are not visible in the left plot, so we zoomed in on one of these poles on the right. For simplicity, we plotted the spectral function in $d=2$. The $d=2$ spectral function is not UV divergent and hence we do not need to renormalize it. It is clearly visible that $\Sigma_\nu$ has positive real part almost everywhere in the complex plane except for small neighborhoods around the poles.
• Figure 5: Pole structure of the 1-loop spectral function $\Sigma_\nu(\tilde{\nu}')$ along the imaginary axis. For simplicity, we plotted the spectral function in $d=2$, which has poles on the imaginary axis at $\pm i(2m+1),\,m\in\mathbb{N}$. This plot was also shown in DiPietro:2023inn, however there the contributions from the time-ordered propagators were missing. The poles of the resummed spectral function are found at the intersection with the constant plane at $-1/\lambda$ (the sign is also different in our case compared to DiPietro:2023inn where there seem to be some issues regarding the positivity of the spectral function, and consequently unitarity). There is one subtlety in this plot: We only show $\Re[\Sigma_\nu]$ and hence intersections in this plot are not necessarily the actual zeros in the complex plane, but the general idea should be clear.